Prior to this lesson students have learned the FOIL method of multiplying two binomials. There has been some criticism of the FOIL method as obscuring the conceptual underpinning of what is happening when multiplying binomials. I provide a derivation of the short-cut. The first method they learned, as discussed in a previous section gives way to the short-cut as you will see shortly. I admonish my students that this initial method is not to be forgotten because the short-cut is hard to apply for multiplying polynomials other than binomials; e.g., (a + b + c)(d + e + f).
I show the derivation using the multiplication of (a + b)(c + d), using the method they have been using:
Each of the terms in the product can be associated with the position of each of the terms in the two binomials as follows:
ac = the product of the first terms in each of the two binomials: a and c
ad = the product of the two outside terms of each of the two binomials: a and d
bc = the product of the two inner terms of each of the two binomials: b and c
bd = the product of the last terms of each of the two binomials: b and d.
This gives way to the mnemonic FOIL which stands for “first, outer, inner, and last” as described above and provides an easier way to do the multiplication. I have found that students find it helpful when learning the factoring of trinomials, often remarking that it is a “reverse FOIL”.
Squaring a binomial provides yet another shortcut, and knowing FOIL helps us quickly get to it.
Again, I remind students that it is to their benefit to know the squares of 11 through 20 and to be able to recognize numbers that are perfect squares. I post these on the board every day, but soon stop doing that. They will find problems on Warm-Ups that are far easier if they have these squares committed to memory, since I don’t allow them to use calculators. The chart I display is shown below:
Warm-Ups.
Problem 1 illustrates the advantage of knowing the squares from 11 to 20. Students may rely on the chart I put on the board, but some students remain oblivious to it, even though I have told them about it. Problem 3 is not a difference of two squares, although some will treat it as such. It is important that students learn to identify what type of factoring will apply. Problem 4 is a distance = distance problem. In this case, the distance the faster car travels in t hours eventually will be 84 miles greater than the distance the slower car travels in the same amount of time. A diagram is helpful to students for such problems. Problem 5 provides a segue to the day’s lesson. In going over the answer, I point out that the problem may also be written as (x + 2)².
Preliminaries. I write Problem 5 of the Warm-Ups with the answer on the board:
“Now I want you to find (x + 3)².” I may have to remind them that it is the same thing as (x + 3) (x + 3).
They do so, and I have them square (x + 4) and (x + 5). I then have them look at the board which has the answers displayed.
“What patterns do you see?” I ask.
All notice right away that the last number is the square of the second term of the binomial. Others see that the coefficient of the middle term of the trinomial is two times the second term.
I then write the following problems on the board, have them do it, and after answers are posted it looks like this:
“What is the pattern you see now?”
Generally they see right away that the middle term is negative, and the last term remains positive.
“Does this suggest a rule that we can use for squaring a binomial?” Students may wrestle with this but ultimately come up with something that sounds close to “The first term is squared and the second term is double the second term; the last term is the second term squared.” This gives me something to work with and I ask them to do one last one: (2x – 5)². They come up with 4x² - 20x + 25.
“This one follows the same pattern as the ones we just did. So let me state the rule formally. It’s a rule that is used to square binomials and can save time.”
Rule for Squaring Binomials. I write the rule on the board using the last example:
Students may get confused when the second term is negative. I remind them that the binomial 2x - 5 is the same thing as 2x + (-5).
Some students ask if they can use the FOIL method for squaring a binomial, rather than the just-learned rule. I answer that they may use whatever method they feel comfortable with and will not take off points if they do it that way on a test or quiz. I point out that the rule saves some time. It is also of value as a way to identify perfect square trinomials, and factor them, which is covered in a future lesson.
Homework. The homework should primarily be problems that require squaring binomials which include terms that are powers such as in Problem 5 above. The assignment should also include other types of multiplication and factoring problems that have been covered in the past few lessons to ensure they are familiar with all these types.